User peter - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T11:31:08Z http://mathoverflow.net/feeds/user/15417 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66462/which-functions-of-one-variable-are-derivatives/66496#66496 Answer by Peter for Which functions of one variable are derivatives ? Peter 2011-05-30T21:46:51Z 2012-12-28T02:34:51Z <p>Take a look a this book by <em>Andrew M. Bruckner</em>: <strong>Differentiation of real functions</strong>.</p> <p>Chapter seven is about <em>The problem of characterizing derivatives</em>.</p> <p>There is a <a href="http://projecteuclid.org/euclid.bams/1183545222" rel="nofollow">review by <em>Daniel Waterman</em></a>.</p> <p>You might also want to take a look at <strong>Homeomorphisms in Analysis</strong> by <em>Goffman</em>, <em>Nishiura</em> and <em>Waterman</em>.</p> http://mathoverflow.net/questions/67182/generalized-gauss-green-theorem Generalized Gauss-Green theorem Peter 2011-06-07T21:12:53Z 2011-06-21T17:50:30Z <p>I am looking for a generalized version of the Gauss-Green theorem also known as the divergence theorem:</p> <p><a href="http://en.wikipedia.org/wiki/Divergence_theorem" rel="nofollow">http://en.wikipedia.org/wiki/Divergence_theorem</a></p> <p>A quick search on <a href="http://www.ams.org/mathscinet/search/publications.html?pg5=TI&amp;s5=gauss+green" rel="nofollow">MathSciNet</a> suggests that there are generalizations for bad domains and nonsmooth functions. However, they seem to rely on heavy machinery and not to be suited for the special case I am interested in.</p> <p>For example, I found this formula on PlanetMath:<code>$$\int_E \mathrm{div} f(x)\, dx = \int_{\partial^* E} \langle \nu_E(x),f(x)\rangle \,d\mathcal H^{n-1}(x)$$</code></p> <p>See <a href="http://planetmath.org/?method=l2h&amp;from=objects&amp;name=GaussGreenTheorem&amp;op=getobj" rel="nofollow">http://planetmath.org/?method=l2h&amp;from=objects&amp;name=GaussGreenTheorem&amp;op=getobj</a> for the details.</p> <hr> <p>Let $\Omega \subset \mathbb{R}^n$ be open and bounded and $f\in C^1(\Omega, \mathbb{R}^n) \cap C^0(\overline\Omega, \mathbb{R}^n)$. </p> <p><strong>Question:</strong> What conditions do we have to impose on $\Omega$ (or $f$) to ensure that the divergence theorem holds true?</p> <hr> <p>To clarify my question: I know that requiring the boundary of $\Omega$ to be piecewise regular is sufficient for the Gauss-Green theorem to be true. I wondered if this condition is also necessary. If so: is the an other "version" of Gauss-Green (e.g. the one cited above) which holds true under weaker conditions and is especially suited for the case of an open and bounded domain</p> http://mathoverflow.net/questions/66240/topological-spaces-uncountable-subsets-and-separability Topological spaces, uncountable subsets and separability Peter 2011-05-27T22:32:11Z 2011-05-28T11:37:27Z <p>Hi, the following is a well known theorem</p> <p><em>Let $M$ be a metric space. If every uncountable subset of $M$ has a limit point, then $M$ is separable.</em></p> <p>Question: Is there a similar result for topological spaces?</p> <p>I have almost no knowledge of topology so I can only hope that this is not trivial.</p> http://mathoverflow.net/questions/67182/generalized-gauss-green-theorem Comment by Peter Peter 2011-06-07T22:09:09Z 2011-06-07T22:09:09Z Wikipedia says that the divergence theorem is also known as Gauss' theorem, Ostrogradsky's theorem, or Gaussâ€“Ostrogradsky theorem. My professor called it the Gauss-Green theorem.